How Long Would It Take This Fan to Stop?

Sometimes I think Dan Meyer does this to me on purpose. He knows I can’t not answer the question. Here is his question: Basically, from this video of a fan, how long would it take to stop?

This isn’t your usual kinematics video — mostly because it involves rotations and not linear motion. So, there are a couple of tricks. You know where to start though, right? Start with Tracker video analysis. And here is the first trick. Make sure you put the origin of your coordinate system in the center of the fan. Like this:

Why do you need to do this? Well, Tracker is going to give you x-y coordinates for some part of the fan in each frame. You don’t really care about x and y. You care about the angular position. If you have the origin at the center of the fan, you can get θ (the angular position) of the fan fairly easily. In fact, Tracker can even do it for you. I didn’t mark all the points of the fan, but here is the first half of the motion.

Yes, I know what you are thinking. That doesn’t look quite right. Well, calculations are sort of stupid in that they just do what you tell them to. If you want the angle the fan has moved using x and y coordinates, they repeat. The calculations don’t automagically take into account how many times the fan went around. You have to do that yourself. Here the angular position is getting smaller and smaller. So, each time it goes around, I can just subtract 2π from the angle and I get something like this:

I could have made this angular data change in Tracker, but if I am going to redo stuff I might as well do it in Python, right? Looking at this data, it looks mostly linear. Aha! But mostly linear is slightly parabolic. Slightly parabolic means I can fit a polynomial function to the data. For me, I will use the polyfit function in PyLab. You could use a spreadsheet if it made you happy. The cool thing is that we don’t even really worry about forces and stuff. But here is the function I get:

But when will it stop? Well, what does “stop” mean? It means that the angular position is no longer changing. In terms of calculus, this means the derivative of θ with respect to time would be zero. That means:

Now, solving this for the time, I get t = 19 seconds. This is the time measured from the t = 0 seconds point in my graph (which is shortly after the fan was turned off). That is your answer. But it seems rather short. Maybe it is OK. It seems the video only shows the fan slowing for 9 seconds. Well, the idea is solid.

Another way to get this

Oh, calculus makes you feel faint? OK. Let’s do something else. If we assume that the angular acceleration is constant, then I can write:

Where α is the angular acceleration and ω is the angular velocity (just so we agree on the terms). In this case, it looks just like the definition of linear acceleration. I could redo the derivation, but you can get to the same thing for the angular position as a function of time (usually called one of the kinematic equations):

Now we have this in a form that is just like our polynomial fit. If you match up the terms, you will see that the term in front of the t2 must be (1/2)α. This means that for this case, the angular acceleration must be:

The polynomial fit also gives the initial angular velocity — in this case it is -9.36 rad/s. So I want to find the time it take for this angular velocity to get to zero, that would be:

There you go. The same time.

I know, they are identical because really they are the same method. I get it.

One More Method

You still aren’t happy, are you? OK, back to the plot from Tracker video. What if I find the slopes of these different apparently straight-looking lines? Here is the first line’s slope.

This makes it sort of look like the rate of change for the angle is constant. These lines look linear, don’t they? Well, look at the slope for this first set. I get an angular velocity of -9.327 rad/s. What if I do the same thing to the last set of points that looks like a line? I get -7.002 rad/s. So, even though these lines might appear to have the same slope, they don’t.

How does the slope change? I have eight sets of data that make lines. Let me plot the slopes of these lines (which would be an approximation for the angular velocity) versus the time in the middle of this data set. Here is what that would look like.

Looks linear, right? The linear function that fits this data has a slope of 0.463 rad/s2 with an intercept of -9.34 rad/s. So, I can write a function for the angular velocity as:

When does it stop? It stops when ω is zero rad/s. If I put in zero for ω, I can solve the time. This gives t = 20.1 seconds. Basically the same value as before (but not quite the same). Why is it different? Well, look at the data. The fit isn’t quite as nice at the parabolic fit from before. This is because I took the data in chunks and found the average angular acceleration.

If you wanted a better fit, you could take maybe 3 data points at a time and find the average angular acceleration. This would give you a better answer, but it would also take a little more effort. Oh, remember that this time is from the start of my data — not the moment the fan was turned off. I wanted to cut out the part with Dan’s hand so it wouldn’t get in the way.

Slight Update

There were some initial claims on twitter that the angular acceleration wasn’t constant. Ok, I could have been wrong. After all, I only looked at the first part of the data. So, skipping the data in the middle, I have a new plot of angular velocity versus time.

This still looks very linear. It did change the slope to 0.398 rad/s2 though. This would change the stop time to 23 seconds. OK, I am mostly happy.

Real Update: Fools Rush In (I am the fool)

Let me chalk this up to “unbridled enthusiasm”. I saw a video and I was excited. In my haste, I didn’t even realize what the real problem was. I am the kid that doesn’t read the whole question on a test.

So, the real problem is that there is another video. In this second video, the fan runs much longer. In fact, the fan does NOT stop in 20 seconds like I said. In this case, the acceleration of the fan is not constant – really, it shouldn’t be. There is obviously some velocity-dependent force on the blades of the fan (air resistance). This means that the angular acceleration is not constant.

But how do you solve a problem with non-constant acceleration? I will just leave this great summary post here: